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The superconducting proximity effect leads to strong modifications of thelocal density of states in diffusive or chaotic cavity Josephson junctions,which displays a phase-dependent energy gap around the Fermi energy. Theso-called minigap of the order of the Thouless energy $E_{\mathrm{Th}}$ isrelated to the inverse dwell time in the diffusive region in the limit$E_{\mathrm{Th}}\ll\Delta$, where $\Delta$ is the superconducting energy gap.In the opposite limit of a large Thouless energy $E_{\mathrm{Th}}\gg\Delta$, asmall new feature has recently attracted attention, namely, the appearance of afurther secondary gap, which is around two orders of magnitude smaller comparedto the usual superconducting gap. It appears in a chaotic cavity just below thesuperconducting gap edge $\Delta$ and vanishes for some value of the phasedifference between the superconductors. We extend previous theory restricted toa normal cavity connected to two superconductors through ballistic contacts toa wider range of contact types. We show that the existence of the secondary gapis not limited to ballistic contacts, but is a more general property of suchsystems. Furthermore, we derive a criterion which directly relates theexistence of a secondary gap to the presence of small transmission eigenvaluesof the contacts. For generic continuous distributions of transmissioneigenvalues of the contacts, no secondary gap exists, although we observe asingular behavior of the density of states at $\Delta$. Finally, we provide asimple one-dimensional scattering model which is able to explain thecharacteristic "smile" shape of the secondary gap.

Secondary "Smile"-gap in the density of states of a diffusive Josephson junction for a wide range of contact types